Anisotropy effects are fundamental to superconductivity. This is true because any non-cubic material that can be grown with sufficient structural perfection should show some anisotropy in its superconductivity. Even cubic materials can show deviation from isotropic models of superconductivity if they have non-spherical Fermi surfaces.[33] Thus, as pointed out by Dalrymple,[50] just about all crystalline superconductors are in principle expected to show some anisotropy effects. Clearly these effects are more important in some materials than in others, and the materials at the high-anisotropy end of the continuum are more interesting for comparison with GIC's.

At the weakly anisotropic end of the continuum are bulk
transition metals like Nb, V, and Tc. These materials are
three-dimensional anisotropic superconductors characterized
by different superconducting coherence lengths in different
crystallographic directions. The Pippard coherence length
*xi*_{0}, is defined to be the approximate
spatial extent of the Cooper pairs. From an
uncertainty-principle-type argument by Pippard,[252] this extent must be
approximately *hbar*/*Delta*p = *hbar*
v_{F} / k_{B} T_{c}. This
relationship shows the connection between the Fermi surface
and superconductivity.

In the Ginzburg-Landau theory of superconductivity (discussed
further in Sections 4.1 and 4.5) a more generalized coherence
length *xi* is defined. This Ginzburg-Landau coherence
length is related to the upper critical field for a bulk type
II superconductor by

where ø_{0} is the superconducting flux
quantum.[252] An
angle-dependent coherence length therefore gives an
anisotropic critical field. When a superconductor has
uniaxial symmetry, two of its three pricipal coherence
lengths are the same. (In the most general case, all three
coherence lengths may be different.) Then the angular
dependence of H_{c2} is given by the simple
expression:[127,175]

where *epsilon* is the critical field anisotropy
parameter of Morris, Coleman and Bhandari,[175], defined by:

Except for this angular dependence of H_{c2}, these
weakly anisotropic superconductors behave for the most part
like isotropic type II superconductors. For example, the
temperature dependence of their critical fields is given by
the usual Werthamer-Helfand-Hohenberg-Maki theory.[54] This theory gives a linear
temperature dependence near T_{c} and saturation at
lower temperatures. Linear behavior of H_{c2} implies
because of Eqn. 1 that *xi* goes as (1 -
t)^{-1/2}.

Besides bulk anisotropic superconductors like some of the transition metals, there is another familiar class of materials with anisotropic superconducting properties. This is the class of superconducting thin films, which were the first anisotropic superconductors to be studied.[250,32] When the thickness of a film is less than the coherence length, the Cooper pairs can only interact with their neighbors in the plane of the film. In this case, the film is commonly referred to as a two-dimensional superconductor because the Cooper pairs only interact in two directions.

As might be expected, the lowering of the effective
dimensionality of a superconductor from three to two
dimensions has important and measurable consequences. These
consequences stem from the fact that the length scale for
superconductivity in the direction perpendicular to the film
is now the film thickness rather than the coherence length.
Most of the expressions that describe thin-film
superconductors can be derived simply by replacing one of the
coherence lengths in Eqn. 1 by the film thickness, plus some
numerical factors. For example, consider the critical fields
of a thin film. When the external field is applied along the
perpendicular to the film, Eqn. 1 still holds. However, for a
field applied in the plane of the film, the critical field
for a thin film is now H_{c2, _|_ ^c} =
*sqrt*3ø_{0} / *pi* *xi* d,
where d is the film thickness.[252] From the (1 -
t)^{-1/2} temperature dependence of *xi* given
above, one can immediately see that H_{c2, _|_ c}
will have a square-root temperature dependence:
H_{c2} proportional to (1 - t)^{1/2}.[250,104]

The square-root temperature dependence is one of the hallmarks of two-dimensional superconductivity. The other is an angular dependence of the critical field given by Tinkham's formula:[250]

where H_{c2}(0°) = H_{c2, || c} and
H_{c2}(90°) = H_{c2, _|_ c}. The angle
*theta* is defined in Figure 4.6. The square-root
temperature dependence and Tinkham's formula for the angular
dependence have been observed many times in thin films. What
is more surprising is that these two-dimensional properties
have also been observed in bulk superconductors which have a
layered structure. Rigorous calculations for a multilayer
structure composed of superconducting and insulating planes
have been performed by Klemm, Luther and Beasley.[131] The KLB calculations show that
a dimensionality crossover is expected to occur at a finite
temperature when the parameter **r** is less than 1.7,[131] where **r** is defined
by

The temperature at which the decoupling of the
superconducting planes occurs occurs is called T^{*}.
T^{*} is approximately defined by *xi*_{||
c}(T^{*}) = d. Usually layered superconductors
show 3D anistropic superconductivity like the bulk transition
metals, but sometimes they show 2D superconductivity like
thin films, and sometimes they even show entirely new
effects. Below, the portion of the work on the upper critical
fields of layered superconductors which is most relevant to
the GIC experiments is briefly reviewed.

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